T0-reflection and injective hulls of fibre spaces

Abstract

We give a characterization of injective (with respect to the class of embeddings) topological fibre spaces using their T 0-reflection, that turns out to be injective itself. We then prove that the existence of an injective hull of (X, f) in the category Top/B of topological fibre spaces is equivalent to the existence of an injective hull of its T 0-reflection (X 0 , f 0) in Top/B 0 (and in the category Top 0 /B 0 of T 0 topological fibre spaces).

On injective objects and existence of injective hulls in 𝑄-TOP/(𝑌, 𝜎)

“…Cagliari and Mantovani studied injective objects and injective hulls in the comma category TOP/𝐵 (cf. [4][5][6]). In particular, they gave a characterization of injective objects (with respect to the class of embeddings in the category TOP of topological spaces) in the comma category TOP/𝐵 (cf.…”

“…In particular, they gave a characterization of injective objects (with respect to the class of embeddings in the category TOP of topological spaces) in the comma category TOP/𝐵 (cf. [5]). In [6], they gave a result related to the existence of an injective hull of an object in the comma category TOP 0 /𝐵 (𝐵 ∈ TOP 0 and TOP 0 is the category of 𝑇 0 -topological spaces).…”

“…It is well known that the category TOP 0 is a reflective subcategory of the category TOP. In [5] (see also [14]), Cagliari and Mantovani considered the reflector 𝜋 : TOP→ TOP 0 and they mentioned that for any topological space 𝑌 , by virtue of the reflector 𝜋, corresponding to each object (𝑋, 𝑓 ) of the comma category TOP/𝑌 we have the object (𝑋 0 , 𝑓 0 ), where 𝑋 0 = 𝜋(𝑋) and 𝑓 0 = 𝜋( 𝑓 ), of the category TOP 0 /𝑌 0 , which is called as the 𝑇 0 -reflection of (𝑋, 𝑓 ). Cagliari and Mantovani [5] gave a characterization of injective objects (with respect to the class of embeddings in TOP) in the comma category TOP/𝑌 with the help of their 𝑇 0reflection.…”

“…In [5] (see also [14]), Cagliari and Mantovani considered the reflector 𝜋 : TOP→ TOP 0 and they mentioned that for any topological space 𝑌 , by virtue of the reflector 𝜋, corresponding to each object (𝑋, 𝑓 ) of the comma category TOP/𝑌 we have the object (𝑋 0 , 𝑓 0 ), where 𝑋 0 = 𝜋(𝑋) and 𝑓 0 = 𝜋( 𝑓 ), of the category TOP 0 /𝑌 0 , which is called as the 𝑇 0 -reflection of (𝑋, 𝑓 ). Cagliari and Mantovani [5] gave a characterization of injective objects (with respect to the class of embeddings in TOP) in the comma category TOP/𝑌 with the help of their 𝑇 0reflection. Cagliari and Mantovani [5] also proved that the existence of an injective hull of (𝑋, 𝑓 ) in the comma category TOP/𝑌 is equivalent to the existence of an injective hull of its 𝑇 0 -reflection (𝑋 0 , 𝑓 0 ) in the comma category TOP/𝑌 0 (and in the comma category TOP 0 /𝑌 0 ).…”

Injectivity and sections

“…[8]) and Scott-continuous functions. ) Recently, new investigations on injective objects have been developed in comma-categories C/B (whose objects are C-morphisms with fixed codomain B) (see [13,14,3,6,7]). "Sliced" injectivity is related to weak factorization systems, a concept used in homotopy theory, particularly for model categories.…”